Warped products provide a rich class of physically significant geometric objects. Warped product construction is an important method to produce a new metric with a base manifold and a fibre. We construct compact base manifolds with a positive scalar curvature which do not admit any non-trivial quasi-Einstein warped product, and non compact complete base manifolds which do not admit any non-triv...

The theory of pseudodifferential operators and Fourier integral operators on compact manifolds is well established and their applications in mathematical physics well known , see for example Hörmander , Duistermaat , Treves 12 . For open (non compact) manifolds this is not the case, and that’s what I would like to focus on in this paper. We are interested in the geometry of the spaces of pseudo...

We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact 4-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds and find conditions assuring that they admit Stein, compact neighborhoods. We base our calculations on Turaev’s shadows suitably “smoothed”; the conditions we find are purely algebraic and combinatorial.

We work primarily in the category of manifolds of bounded geometry. The objects are manifolds with bounds on the curvature tensor, its derivatives, and on the injectivity radius. The morphisms are diffeomorphisms of bounded distortion. We think of these manifolds as having a chosen bounded distortion class of metrics. Unless otherwise stated, all manifolds in the paper are assumed of this type....

We construct the non-compact Calabi-Yau manifolds interpreted as the complex line bundles over the Hermitian symmetric spaces. These manifolds are the various generalizations of the complex line bundle over CP. Imposing an F-term constraint on the line bundle over CP, we obtain the line bundle over the complex quadric surface Q. On the other hand, when we promote the U(1) gauge symmetry in CP t...

The base space of an algebraically completely integrable Hamiltonian system acquires a rather special differential-geometric structure which plays an important role in modern physical theories such as Seiberg-Witten theory. This structure was formalised by D. Freed [1] as a Special Kähler manifold. In that paper he conjectured that there are no compact special Kähler manifolds other than flat o...

An origami manifold is a manifold equipped with a closed 2-form which is symplectic except on a hypersurface where it is like the pullback of a symplectic form by a folding map and its kernel fibrates with oriented circle fibers over a compact base. We can move back and forth between origami and symplectic manifolds using cutting (unfolding) and radial blow-up (folding), modulo compatibility co...

Using techniques of supersymmetric gauge theories, we present the Ricci-flat metrics on non-compact Kähler manifolds whose conical singularity is repaired by the Hermitian symmetric space. These manifolds can be identified as the complex line bundles over the Hermitian symmetric spaces. Each of the metrics contains a resolution parameter which controls the size of these base manifolds, and the ...

In this paper we will show that any complete manifold of nonnegative curvature has a flat soul provided it has curvature going to zero at infinity. We also show some similar results about manifolds with bounded curvature at infinity. To establish these theorems we will prove some rigidity results for Riemannian submersions, eg., any Riemannian submersion with complete flat total space and compa...

We find sufficient conditions for the absence of harmonic L spinors on spin manifolds constructed as cone bundles over a compact Kähler base. These conditions are fulfilled for certain perturbations of the Euclidean metric, and also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vişinescu and the second author.